In this dissertation, we introduce a family of fully discrete finite difference time-domain (FDTD) methods for Maxwell’s equations in linear and nonlinear materials. Onecategory of methods is constructed using multiscale techniques involving operator splittings. We present the sequential splitting scheme, the Strang Marchuk splitting scheme,the weighted sequential splitting scheme including...
We generalize overpartition rank and crank generating functions to obtain k-fold variants, and give a combinatorial interpretation for each. The k-fold crank generating function is interpreted by extending the first and second residual cranks to a natural infinite family. The k-fold rank generating functions generate two families of buffered Frobenius...
In this note, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our extended technique could be a useful tool to prove congruences for certain types of combinatorial...
In this thesis I will look at a definition of computable randomness from Algorithmic Information Theory as defined by Andre Nies through the lens of Computable Analaysis asdefined by Klaus Weihrauch. I will show that despite the fact that these two paradigmsgenerate distinct classes of computable supermartingales, the class of...
The fact that measuring a quantum system reduces it to apparently classical behavior, eliminating the interference patterns that are a hallmark of quantumness, cries out for an explanation. That explanation has been provided by the recognition of decoherence,whereby the interference is destroyed by the very interaction that acquires information.We begin...
Markov chains have long been used to sample from probability distributions and simulate dynamical systems. In both cases we would like to know how long it takes for the chain's distribution to converge to within varepsilon of the stationary distribution in total variation distance; the answer to this is, called...
In this study, I seek to examine undergraduate STEM majors’ beliefs about and attitude towards mistakes in the context of counting. This is a particularly fruitful setting for such an investigation both because combinatorics is widely applicable to various fields such as physics, biology, chemistry, and computer science (Kapur, 1970),...
This dissertation investigates the structure and topological properties of cyclicallypresented groups. First, a family of groups called groups of type Z is considered. Withfew exceptions, the finiteness, asphericity, fixed point, and 3-manifold spine problemsare solved. Most groups of type Z have a central element of infinite order fixed by theshift....
The goal of this paper is to classify linear operators with octonionic coefficients and octonionic variables. While building up to the octonions we also classify linear operators over the quaternions and show how to relate the linear operators over the quaternions and octonions to matrices. We also construct a basis...